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How did physicists interpret the physical significance of retarded potential before the advent of special relativity? , Did it account for the time taken by electromagnetic forces to propagate through aether or was it simply accepted as a superset of existing formulas with those being the trivial cases of steady currents and static charge?

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  • $\begingroup$ The speed of light was known before special relativity. The whole point of retarded potentials was to account for that fact. $\endgroup$ – Jon Custer Feb 27 '20 at 15:33
  • $\begingroup$ Can you please elaborate? $\endgroup$ – Kutsit Feb 27 '20 at 15:40
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Although I can't give an accurate in-depth view of what 19th century physicists thought about the results of Electromagnetic Theory, they most likely interpreted the retarded potentials similar to what we interpret nowadays: that "electromagnetic news" travel at the speed of light.

It was already known at Maxwell's time that the speed of light was finite and roughly equal to $3 \times 10^8$ m/s. That light had a finite speed was not a problem to 19th century physicists (it was a fact!), their problem was in which reference frame was the speed of light measured?. And EM theory doesn't explain about the reference frame. That's why physicists tested for the aether for so long, they only had experience with waves that propagates through a material medium. Only Special Relativity tells you about the reference frames (and fixes the inconsistencies) because the problem was actually about Mechanics and the old newtonian views, and not about EM.

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